What is the value of \(\frac{a^{2}~+~ac}{a^{2}c~-~c^{3}}-\frac{a^{2}~-~c^{2}}{a^{2}c~+~2ac^2~+~c^3}-\frac{2c}{a^2~-~c^2}+\frac{3}{a~+~c}\) ?
Understanding the Algebraic Expression Simplification
We need to simplify the given algebraic expression. This involves factoring polynomials, finding a common denominator, and combining terms. Let's break down the expression term by term.
The expression is:
\(E = \frac{a^{2}~+~ac}{a^{2}c~-~c^{3}}-\frac{a^{2}~-~c^{2}}{a^{2}c~+~2ac^2~+~c^3}-\frac{2c}{a^2~-~c^2}+\frac{3}{a~+~c}\)
Simplifying Each Term
Let's simplify each fraction in the expression by factoring the numerator and the denominator.
- First term: \(\frac{a^{2}~+~ac}{a^{2}c~-~c^{3}}\)
- Numerator: \(a^{2} + ac = a(a+c)\)
- Denominator: \(a^{2}c - c^{3} = c(a^{2} - c^{2}) = c(a-c)(a+c)\)
- Simplified first term: \(\frac{a(a+c)}{c(a-c)(a+c)} = \frac{a}{c(a-c)}\) (assuming \(a+c \neq 0\))
- Second term: \(\frac{a^{2}~-~c^{2}}{a^{2}c~+~2ac^2~+~c^3}\)
- Numerator: \(a^{2} - c^{2} = (a-c)(a+c)\)
- Denominator: \(a^{2}c + 2ac^{2} + c^{3} = c(a^{2} + 2ac + c^{2}) = c(a+c)^2\)
- Simplified second term: \(\frac{(a-c)(a+c)}{c(a+c)^2} = \frac{a-c}{c(a+c)}\) (assuming \(a+c \neq 0\))
- Third term: \(\frac{2c}{a^2~-~c^2}\)
- Numerator: \(2c\)
- Denominator: \(a^2 - c^2 = (a-c)(a+c)\)
- Simplified third term: \(\frac{2c}{(a-c)(a+c)}\)
- Fourth term: \(\frac{3}{a~+~c}\) (already in simplest form)
Substitute the simplified terms back into the expression:
\(E = \frac{a}{c(a-c)} - \frac{a-c}{c(a+c)} - \frac{2c}{(a-c)(a+c)} + \frac{3}{a~+~c}\)
Finding a Common Denominator
The denominators are \(c(a-c)\), \(c(a+c)\), \((a-c)(a+c)\), and \((a+c)\). The least common denominator (LCD) is \(c(a-c)(a+c)\).
Rewrite each term with the LCD:
- \(\frac{a}{c(a-c)} = \frac{a \cdot (a+c)}{c(a-c) \cdot (a+c)} = \frac{a(a+c)}{c(a-c)(a+c)}\)
- \(\frac{a-c}{c(a+c)} = \frac{(a-c) \cdot (a-c)}{c(a+c) \cdot (a-c)} = \frac{(a-c)^2}{c(a-c)(a+c)}\)
- \(\frac{2c}{(a-c)(a+c)} = \frac{2c \cdot c}{(a-c)(a+c) \cdot c} = \frac{2c^2}{c(a-c)(a+c)}\)
- \(\frac{3}{a+c} = \frac{3 \cdot c(a-c)}{(a+c) \cdot c(a-c)} = \frac{3c(a-c)}{c(a-c)(a+c)}\)
Now, combine the terms with the common denominator:
\(E = \frac{a(a+c) - (a-c)^2 - 2c^2 + 3c(a-c)}{c(a-c)(a+c)}\)
Expanding and Simplifying the Numerator
Expand the terms in the numerator:
- \(a(a+c) = a^2 + ac\)
- \((a-c)^2 = a^2 - 2ac + c^2\)
- \(2c^2\)
- \(3c(a-c) = 3ac - 3c^2\)
Substitute these back into the numerator expression:
Numerator \( = (a^2 + ac) - (a^2 - 2ac + c^2) - 2c^2 + (3ac - 3c^2)\)
Remove the parentheses, remembering to change signs for the terms inside the second parenthesis (because of the minus sign in front):
Numerator \( = a^2 + ac - a^2 + 2ac - c^2 - 2c^2 + 3ac - 3c^2\)
Group like terms:
Numerator \( = (a^2 - a^2) + (ac + 2ac + 3ac) + (-c^2 - 2c^2 - 3c^2)\)
Combine like terms:
Numerator \( = 0a^2 + 6ac - 6c^2\)
Numerator \( = 6ac - 6c^2\)
Factor the numerator:
Numerator \( = 6c(a-c)\)
Final Simplification
Now substitute the simplified numerator back into the expression:
\(E = \frac{6c(a-c)}{c(a-c)(a+c)}\)
Assuming \(c \neq 0\) and \(a-c \neq 0\), we can cancel out the common factor \(c(a-c)\) from the numerator and the denominator:
\(E = \frac{6}{a+c}\)
This is the simplified value of the expression.
| Step | Operation | Result |
|---|---|---|
| 1 | Factor first term | \(\frac{a}{c(a-c)}\) |
| 2 | Factor second term | \(\frac{a-c}{c(a+c)}\) |
| 3 | Factor third term denominator | \(\frac{2c}{(a-c)(a+c)}\) |
| 4 | Find LCD | \(c(a-c)(a+c)\) |
| 5 | Rewrite with LCD and combine numerators | \(\frac{a(a+c) - (a-c)^2 - 2c^2 + 3c(a-c)}{c(a-c)(a+c)}\) |
| 6 | Simplify numerator | \(6c(a-c)\) |
| 7 | Final simplification | \(\frac{6}{a+c}\) |
Conclusion
The value of the given algebraic expression is \(\frac{6}{a+c}\).
Revision Table: Key Concepts in Algebraic Simplification
| Concept | Description | Importance |
|---|---|---|
| Factoring Polynomials | Breaking down a polynomial into simpler expressions (factors) that multiply together to give the original polynomial. E.g., \(a^2 - c^2 = (a-c)(a+c)\). | Essential for simplifying fractions and finding common denominators. |
| Least Common Denominator (LCD) | The smallest multiple that is common to all denominators in a set of fractions. For algebraic fractions, it's the product of the highest powers of all prime factors in the denominators. | Necessary for adding or subtracting fractions. |
| Combining Algebraic Fractions | To add or subtract fractions, they must have a common denominator. Once they do, combine the numerators over the common denominator. | Allows performing addition/subtraction operations on rational expressions. |
| Simplifying Fractions | Canceling common factors from the numerator and denominator. This reduces the fraction to its simplest form. | Makes expressions easier to work with and understand. |
Additional Information: Working with Rational Expressions
Rational expressions are fractions where the numerator and denominator are polynomials. Simplifying, adding, subtracting, multiplying, and dividing rational expressions follow rules similar to those for numerical fractions, but they involve polynomial operations like factoring.
- When simplifying, always factor completely before canceling common factors. Remember that you can only cancel factors, not individual terms. For example, in \(\frac{a+b}{a}\), you cannot cancel \(a\).
- When adding or subtracting, finding the LCD is crucial. Multiply the numerator and denominator of each fraction by the factors needed to make the denominator the LCD.
- Pay attention to restrictions on the variables. The simplified expression is equivalent to the original expression only for values of the variables for which the original expression is defined (i.e., the denominators are non-zero). In this problem, \(c \neq 0\), \(a \neq c\), and \(a \neq -c\).
Similar Questions
What is the square root of 4x 4 + 8x 3 - 4x + 1?
If x + 1/x = 8, then what is x 2 + 1/x 2 equal to?
If a + b = 5 and ab = 6, then what is the value of a 3+ b 3?
If \(x=a+b+\frac{(a-b)^2}{4 a+4 b}\) and \(y=\frac{a+b}{4}+\frac{a b}{a+b}\) then what is the value of (x - a)2 - (y - b)2 ?
The quotient of 8x 3– y 3when divided by 2xy + 4x 2+ y 2is
If 2x - 3y - 7 = 0, then what is the value of 8x 3 - 36x 2y + 54xy 2 - 27y 3 - 340 ?
If \(A + B = \rm \frac{x^2 - 8}{x + 2} \ \ and \ A - B = \frac{-x^2 + 2x + 4}{x + 2}\) then what is B equal to ?
If \(96 - 64a^3 + \frac{8}{a^6} - \frac{48}{a^3 } - t^3 = 0\) then what is a 2t + 4a 3 equal to ?
The sum of all possible products taken two at a time out of the numbers \(\pm 1, \pm 2, \pm 3, \pm 4, \pm 5\) is
If \(\left( {{x^8} + \frac{1}{{{x^8}}}} \right) = 47\) , what is the value of \(\left( {{x^6} + \frac{1}{{{x^6}}}} \right)?\)
Important Questions from Identities
If \(2x + { {1} \over 3x}= 5, x ≠ 0\) , then what is the value of \(27x^3+{{1} \over 8x^3}\) ?
If x 2 + 4y 2 = 40, xy = 6 and x > 2y then the value of x - 2y is:
- \(\dfrac{(5.17-2.19)^2-(5.17+2.19)^2}{11.3223}\)
If a(a + b + c) 2= 1792; b(a + b + c) 2= 1536; c(a + b + c) 2= 768 then what will be the value of a?
If a + b + c = 5, a 2+ b 2+ c 2= 27, a 3 + b 3 + c 3 = 125, then the value of \(\frac{abc}{5}\) is:
